Theorem colperpexlem2 25623

Description: Lemma for colperpex 25625. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.)

Hypotheses

Ref	Expression
colperpex.p	|- P = ( Base ` G )
colperpex.d	|- .- = ( dist ` G )
colperpex.i	|- I = (Itv ` G )
colperpex.l	|- L = (LineG ` G )
colperpex.g	|- ( ph -> G e. TarskiG )
colperpexlem.s	|- S = (pInvG ` G )
colperpexlem.m	|- M = ( S ` A )
colperpexlem.n	|- N = ( S ` B )
colperpexlem.k	|- K = ( S ` Q )
colperpexlem.a	|- ( ph -> A e. P )
colperpexlem.b	|- ( ph -> B e. P )
colperpexlem.c	|- ( ph -> C e. P )
colperpexlem.q	|- ( ph -> Q e. P )
colperpexlem.1	|- ( ph -> <" A B C "> e. (∟G ` G ) )
colperpexlem.2	|- ( ph -> ( K ` ( M ` C ) ) = ( N ` C ) )
colperpexlem2.e	|- ( ph -> B =/= C )

Assertion

Ref	Expression
colperpexlem2	|- ( ph -> A =/= Q )

Proof of Theorem colperpexlem2

Step	Hyp	Ref	Expression
1		colperpexlem2.e	. . 3 |- ( ph -> B =/= C )
2		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ A = Q ) -> A = Q )
3	2	fveq2d 6195	. . . . . . . . 9 |- ( ( ph /\ A = Q ) -> ( S ` A ) = ( S ` Q ) )
4		colperpexlem.m	. . . . . . . . 9 |- M = ( S ` A )
5		colperpexlem.k	. . . . . . . . 9 |- K = ( S ` Q )
6	3, 4, 5	3eqtr4g 2681	. . . . . . . 8 |- ( ( ph /\ A = Q ) -> M = K )
7	6	fveq1d 6193	. . . . . . 7 |- ( ( ph /\ A = Q ) -> ( M ` ( M ` C ) ) = ( K ` ( M ` C ) ) )
8		colperpex.p	. . . . . . . . 9 |- P = ( Base ` G )
9		colperpex.d	. . . . . . . . 9 |- .- = ( dist ` G )
10		colperpex.i	. . . . . . . . 9 |- I = (Itv ` G )
11		colperpex.l	. . . . . . . . 9 |- L = (LineG ` G )
12		colperpexlem.s	. . . . . . . . 9 |- S = (pInvG ` G )
13		colperpex.g	. . . . . . . . 9 |- ( ph -> G e. TarskiG )
14		colperpexlem.a	. . . . . . . . 9 |- ( ph -> A e. P )
15		colperpexlem.c	. . . . . . . . 9 |- ( ph -> C e. P )
16	8, 9, 10, 11, 12, 13, 14, 4, 15	mirmir 25557	. . . . . . . 8 |- ( ph -> ( M ` ( M ` C ) ) = C )
17	16	adantr 481	. . . . . . 7 |- ( ( ph /\ A = Q ) -> ( M ` ( M ` C ) ) = C )
18		colperpexlem.2	. . . . . . . 8 |- ( ph -> ( K ` ( M ` C ) ) = ( N ` C ) )
19	18	adantr 481	. . . . . . 7 |- ( ( ph /\ A = Q ) -> ( K ` ( M ` C ) ) = ( N ` C ) )
20	7, 17, 19	3eqtr3rd 2665	. . . . . 6 |- ( ( ph /\ A = Q ) -> ( N ` C ) = C )
21		colperpexlem.b	. . . . . . . 8 |- ( ph -> B e. P )
22		colperpexlem.n	. . . . . . . 8 |- N = ( S ` B )
23	8, 9, 10, 11, 12, 13, 21, 22, 15	mirinv 25561	. . . . . . 7 |- ( ph -> ( ( N ` C ) = C <-> B = C ) )
24	23	adantr 481	. . . . . 6 |- ( ( ph /\ A = Q ) -> ( ( N ` C ) = C <-> B = C ) )
25	20, 24	mpbid 222	. . . . 5 |- ( ( ph /\ A = Q ) -> B = C )
26	25	ex 450	. . . 4 |- ( ph -> ( A = Q -> B = C ) )
27	26	necon3ad 2807	. . 3 |- ( ph -> ( B =/= C -> -. A = Q ) )
28	1, 27	mpd 15	. 2 |- ( ph -> -. A = Q )
29	28	neqned 2801	1 |- ( ph -> A =/= Q )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888   <"cs3 13587   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336  pInvGcmir 25547  ∟Gcrag 25588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352  df-mir 25548

This theorem is referenced by:  colperpexlem3  25624